Question

Evaluate each of the following integrals in the easiest way you can.

$\mathbf{\oint }\left(2ydx-3xdy\right)$around the square bounded by $\mathbf{x}\mathbf{=}\mathbf{3}\mathbf{,}\mathbf{x}\mathbf{=}\mathbf{5}\mathbf{,}\mathbf{y}\mathbf{=}\mathbf{}\mathbf{1}\mathbf{}\mathbf{andy}\mathbf{=}\mathbf{3}$

Short answer

The solution to this problem is /=-20.

Step-by-step solution

Given Information.

The given information is$\oint \left(2ydx-3xdy\right)$

Definition of Green’s Theorem.

The Green's theorem connects a line integral around a simple closed curve C to a double integral over the plane region D circumscribed by C in vector calculus. Stokes' theorem has a two-dimensional special case.

Find the solution.

Write the values of P and Q.

P = 2y

Q = - 3x

Find the difference of their differentiation

$$\begin{gathered} \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}=-3-2 \\ =-5 \end{gathered}$$

The integral is in the form mentioned below.

$$\begin{gathered} I=-5{\int }_1^3{\int }_3^{5}dxdy \\ =-5\times 4 \\ =-20 \end{gathered}$$

Hence, the solution to this problem is / = - 20.